Short answers, pulled from the story.
Sati is a Pali term literally meaning "memory" or "retention," commonly translated into English as mindfulness. In Buddhist practice it refers to maintaining a lucid awareness of bodily and mental phenomena, calling to mind wholesome states such as the seven factors of enlightenment and the Noble Eightfold Path.
Thomas William Rhys Davids first translated sati as "mindfulness" in 1881, in the phrase sammā-sati, which he rendered as "Right Mindfulness; the active, watchful mind." Before that, Daniel John Gogerly had translated sammā-sati as "correct meditation" in 1845.
"Bare attention" is a popular contemporary gloss for sati, introduced by Nyanaponika Thera, but scholars including Georges Dreyfus, Robert H. Sharf, and Bhikkhu Bodhi argue it is an incomplete rendering. Classical Buddhist practice pairs sati with sampajanna (clear comprehension) and atappa (ardency), and traditional teaching is oriented toward acquiring "correct view" and ethical discernment, not a nonjudgmental present-centered awareness.
The Sanskrit equivalent is smriti, written as स्मृति. It literally means "that which is remembered" and was translated into Tibetan as trenpa (dran pa) and into Chinese as nian, a character composed of the elements for "now" and "heart/mind."
Correct mindfulness, called sammā-sati in Pali and samyak-smriti in Sanskrit, is the seventh element of the Noble Eightfold Path. Sati is also the first factor of the Seven Factors of Enlightenment.
Anapanasati means mindfulness of breathing, combining sati (mindfulness) with anapana (inhalation and exhalation). According to tradition the Buddha taught it in the Anapanasati Sutta (MN 118), and it is now practiced across Tibetan, Zen, Tiantai, and Theravada schools as well as western-based mindfulness programs.